The mode of a discrete random variable X with pmf p(x) is that value x* for which p(x) is largest (the most probable x value).

a. Let . By considering the ratio

, show that b(x; n, p) increases with x as long as . Conclude that the mode x* is the integer satisfying .

b. Show that if X has a Poisson distribution with parameter m, the mode is the largest integer less than m. If m is an integer, show that both and m are modes.